Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces

On a real hypersurface M in a complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the k-th g-Tanaka–Webster connection. Associated to g-Tanaka–Webster connection we can define a differential operator of first order. We classify real hypersurfaces such that both the Lie derivative and this differential operator, either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the structure Jacobi operator of M.